Secret Sharing, Rank Inequalities, and Information Inequalities

Abstract

Beimel and Orlov proved that all information inequalities on four or five variables, together with all information inequalities on more than five variables that are known to date, provide lower bounds on the size of the shares in secret sharing schemes that are at most linear on the number of participants. We present here another two negative results about the power of information inequalities in the search for lower bounds in secret sharing. First, we prove that all information inequalities on a bounded number of variables can only provide lower bounds that are polynomial on the number of participants. And second, we prove that the rank inequalities that are derived from the existence of two common informations can provide only lower bounds that are at most cubic in the number of participants.

On the Information Ratio of Non-perfect Secret Sharing Schemes

“…Such bounds can be found by using the entropy function, a method initiated by Karnin et al [37] and Capocelli et al [13]. On the basis of the connections between information theory, matroid theory, and secret sharing found by Fujishige [31,32], Brickell and Davenport [12], and Csirmaz [22], matroids and polymatroids have appeared to be a powerful tool, as it can be seen from several recent works [2,4,5,40,41].…”

“…On the basis of the connection between Shannon entropy and polymatroids that was discovered by Fujishige [31,32] and is described here in Theorem 5.3, lower bounds on the information ratio of perfect secret sharing schemes can be obtained by using linear programming [22,40,48]. Nevertheless, several limitations on this approach have been found [5,22,41]. In this section, we discuss the extension of this method to non-perfect secret sharing.…”

“…As a consequence, the infimum in (3) is a minimum and, moreover, κ(Φ) has a rational value if Φ is a rational access function. This linear programming approach has been used in [22,28,48] and many other works to find lower bounds on the optimal information ratio of perfect secret sharing schemes, but important limitations to this method have been found [5,22,41]. The first of those limitation results [22, Theorem 3.5] can be generalized to the non-perfect case by using the same idea in the proof.…”

“…By Proposition 5.7, this is not possible by using the polymatroid technique, which is based only on the Shannon information inequalities. Better lower bounds could be found by using non-Shannon information inequalities, but limitations on this approach for perfect secret sharing schemes have been found [5,41].…”

Linear secret sharing and the automatic search of linear rank inequalities

“…Linear secret sharing is an important topic in cryptography and multiparty computation. It is known that one can use linear rank inequalities (which are the linear inequalities satisfied by linear polymatroids) to compute lower bounds on the linear optimal information ratio of different access structures [5], [13]. Actually, it seems that it is the only way of obtaining nontrivial lower bounds for the optimal information ratio of general access structures.…”